What's your routing ORD-NRT-SIN? You should have plenty of time...the UA gates are very close to the RCC, and with 12+ shower facilities, there's rarely a long wait. You just have to gauge it for your particular arrival, but after 12+ hours ORD-NRT, a shower feels great before going onward to SIN (another 7+ hours!).

It sounds like you're not upgraded...if you can come across a Sweet Spot, that'd really make your trip. It's a lot of flying.

When you arrive in SIN, it's late...and there's usually a long taxi queue; walk across the street and hire a limo...I think it's only US$10 more, and well worth it since you get your ride immediately instead of waiting 10-20 minutes.